528 research outputs found

### Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque

Mercury is entrapped in a 3:2 resonance: it rotates on its axis three times
for every two revolutions it makes around the Sun. It is generally accepted
that this is due to the large value of the eccentricity of its orbit. However,
the mathematical model originally introduced to study its spin-orbit evolution
proved not to be entirely convincing, because of the expression commonly used
for the tidal torque. Only recently, in a series of papers mainly by Efroimsky
and Makarov, a different model for the tidal torque has been proposed, which
has the advantages of being more realistic, and of providing a higher
probability of capture in the 3:2 resonance with respect to the previous
models. On the other hand, a drawback of the model is that the function
describing the tidal torque is not smooth and consists of a superposition of
kinks, so that both analytical and numerical computations turn out to be rather
delicate: indeed, standard perturbation theory based on power series expansion
cannot be applied and the implementation of a fast algorithm to integrate the
equations of motion numerically requires a high degree of care. In this paper,
we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by
the realistic model: In particular, we present numerical and analytical results
about the nature of the librations of Mercury's spin in the 3:2 resonance. The
results provide evidence that the librations are quasi-periodic in time.Comment: 32 pages, 8 figures, 5 table

### Basins of attraction in forced systems with time-varying dissipation

We consider dissipative periodically forced systems and investigate cases in
which having information as to how the system behaves for constant dissipation
may be used when dissipation varies in time before settling at a constant final
value. First, we consider situations where one is interested in the basins of
attraction for damping coefficients varying linearly between two given values
over many different time intervals: we outline a method to reduce the
computation time required to estimate numerically the relative areas of the
basins and discuss its range of applicability. Second, we observe that
sometimes very slight changes in the time interval may produce abrupt large
variations in the relative areas of the basins of attraction of the surviving
attractors: we show how comparing the contracted phase space at a time after
the final value of dissipation has been reached with the basins of attraction
corresponding to that value of constant dissipation can explain the presence of
such variations. Both procedures are illustrated by application to a pendulum
with periodically oscillating support.Comment: 16 pages, 13 figures, 7 table

### Quasi-periodic attractors, Borel summability and the Bryuno condition for strongly dissipative systems

We consider a class of ordinary differential equations describing
one-dimensional analytic systems with a quasi-periodic forcing term and in the
presence of damping. In the limit of large damping, under some generic
non-degeneracy condition on the force, there are quasi-periodic solutions which
have the same frequency vector as the forcing term. We prove that such
solutions are Borel summable at the origin when the frequency vector is either
any one-dimensional number or a two-dimensional vector such that the ratio of
its components is an irrational number of constant type. In the first case the
proof given simplifies that provided in a previous work of ours. We also show
that in any dimension $d$, for the existence of a quasi-periodic solution with
the same frequency vector as the forcing term, the standard Diophantine
condition can be weakened into the Bryuno condition. In all cases, under a
suitable positivity condition, the quasi-periodic solution is proved to
describe a local attractor.Comment: 10 page

### Direct loss-based seismic design of reinforced concrete frame and wall structures

This paper presents a procedure to design reinforced concrete (RC) buildings to achieve an acceptable target level of earthquake-induced loss (e.g., deaths, dollars, downtime) under a site-specific hazard profile. The procedure is called “direct” since the target loss level is specified at the first step of the process, and virtually no iteration is required. The procedure is based on a simplified loss assessment involving a surrogate model for the seismic demand (i.e., probability distribution of peak horizontal deformation given ground-motion intensity) and simplified loss models for direct and indirect losses. For an arbitrarily-selected target loss level and structural geometry, the procedure provides the force-displacement curve of the corresponding equivalent single degree of freedom system. The principles of displacement-based design are adopted to provide member detailings (beams, columns, walls) consistent with such force-displacement curve. The procedure is applied to 16 realistic RC case studies with a lateral resisting system composed of frames in one direction and cantilever walls in the perpendicular one. They show different geometries, hazard profiles, and target values of direct economic expected annual loss. A benchmark loss estimation is obtained using cloud-based non-linear time-history analyses of multi-degree of freedom models. The procedure is conservative since the benchmark loss levels are always smaller than the targets. Such discrepancy is within 10% for 12 out of 32 case studies, between 10% and 20% for 13, between 20% and 31% for the remaining six. Therefore, the proposed procedure is deemed dependable for preliminary design

### Stable dynamics in forced systems with sufficiently high/low forcing frequency

We consider a class of parametrically forced Hamiltonian systems with one-and-a-half degrees of freedom and study the stability of the dynamics when the frequency of the forcing is relatively high or low. We show that, provided the frequency of the forcing is sufficiently high, KAM theorem may be applied even when the forcing amplitude is far away from the perturbation regime. A similar result is obtained for sufficiently low frequency forcing, but in that case we need the amplitude of the forcing to be not too large; however we are still able to consider amplitudes of the forcing which are outside of the perturbation regime. Our results are illustrated by means of numerical simulations for the system of a forced cubic oscillator. In addition, we find numerically that the dynamics are stable even when the forcing amplitude is very large (beyond the range of validity of the analytical results), provided the frequency of the forcing is taken correspondingly low

### Periodic and quasi-periodic attractors for the spin-orbit evolution of Mercury with a realistic tidal torque

In this paper, we make a detailed study of the spin-orbit dynamics of Mercury, as predicted by the realistic model which has been recently introduced in a series of papers mainly by Efroimsky and Makarov. We present numerical and analytical results concerning the nature of the librations of Mercury’s spin in the 3:2 resonance. The results provide evidence that the librations are quasi-periodic in time, consisting of a slow oscillation, with an amplitude of order of arcminutes, superimposed on the 88-day libration. This contrasts with recent astronomical observations and hence suggests that the 3:2 resonance in which Mercury has been trapped might have been originally described by a large-amplitude quasi-periodic libration which, only at a later stage, with the formation of a molten core, evolved into the small-amplitude libration which is observed nowadays

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